AbstractsMathematics

In this dissertation we construct a homology theory on the category of submersions which generalizes the homology of the base space with coefficients in the homology of the fiber as given by the E²-terms of the Serre spectral sequence of a fiber bundle. The main motivation for this new homology theory is the fact that it permits a generalization of the Serre spectral sequence to arbitrary submersions. The homology theory in question is first defined on a category of combinatorial objects called simplicial bundles which at once generalize the notion of fiber bundles (over polyhedra) and simplicial complexes. We next enlarge the category of submersions to include all direct limits of simplicial bundles and extend the homology functor by a category-theoretic construction. The resultant theory is shown to satisfy axioms of Eilenberg-Steenrod type, and we prove a uniqueness theorem.